Risk transference constraints in optimal reinsurance

Abstract

This paper deals with the optimal reinsurance problem and involves the goals of both insurer and reinsurer. An important novelty may be the incorporation of the background risk that the reinsurer uses in order to diversify (or hedge) the risk ceded by the insurer. Accordingly, general methods to prevent the reinsurer moral hazard must be extended, and a new constraint must be satisfied by the selected reinsurance contract, namely, "the reinsurer increment of risk must be lower than the contract premium". Simultaneously, since the contract must be attractive to the insurer too, "the contract premium must be lower than the insurer risk reduction". Integrating both ideas, "the contract premium must be higher than the reinsurer risk growth and lower than the insurer risk mitigation". Bearing in mind both requirements, that is, the protection against the moral hazard and the spread containing the contract premium, the optimal reinsurance problem is studied under very general conditions about the involved risk measures and premium principles, general solutions are provided, and a practical illustrative example is presented.